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A Comprehensive Guide on Mode

Published by at September 21st, 2021 , Revised On July 5, 2022

What is mode?

Mode or modal value is the most frequent value in a given set of values. The values, collectively, form what is called a dataset. The number most repeated in a dataset is said to be the mode of that dataset.

How is mode calculated?

The method for calculating mode depends on the kind of data. It can be either grouped or ungrouped data. 

Calculating mode for ungrouped data

Ungrouped datasets contain a couple of values that need not be arranged or following some specific pattern. Identifying mode for ungrouped data is simpler, for it only requires finding out the value that occurs the greatest number of times in the dataset; that value will be the mode. 

Ungrouped datasets look something like this: {2, 5, 2, 6, 7, 3, 2,…}. There’s no pattern to the arrangement of the values in such datasets. In this example, though, the mode is 2 since it is repeated 3 times.

Calculating mode for grouped data

Grouped data is a little trickier to deal with than ungrouped data. In this kind of dataset, the values are arranged in two columns or two rows. One represents observed values and the other represents frequencies that respond to those values. For instance, the following is a grouped dataset:

Class levels No. of students who obtained 99 marks in a final exam
2 – 4 3
4 – 6 4
6 – 8 2
8 – 10 1

The formula for calculating mode for such grouped datasets is:

Mode = Z = L + (f1−f0) ÷ (2f1−f0−f2) × h

Where, 

  • L = the lowest value of the modal class
  • h = the size of the class interval
  • f1 = the frequency of the modal class
  • f0 = the frequency of the class interval present before modal class 
  • f2 = the frequency of the class interval present after model class

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Some Important Terms for Mode of Grouped Data

1. Modal class: In grouped data, the group of numbers of the ‘interval’ that contains the mode or the greatest number of times a value occurs, is called the modal class. 

2. Class interval: Grouped datasets themselves can be of further two forms. One is where data is grouped such that there are intervals in between numbers, such as in the dataset table given above. Similarly, 2 to 10 is also an example of an interval.

However, if there are no intervals, data is simply grouped into two columns or rows where one contains observed, independent variables and the other contains frequencies or dependent variables corresponding to the observations. 

For instance, suppose 10 children visit their street park 4 times and 5 children visit it 2 times. In such grouped data (with no intervals), the number of children is independent variables, and their corresponding frequencies are the dependent variables.

3. The lower limit of the modal class: Since a class, interval has two values—one where the interval beings from and one where it ends at—the lower or the starting one are called the lower limit of that class interval. 

Examples to Calculate Mode for Grouped Data

First, let’s consider grouped data with a class interval. Taking the same values as the one mentioned above,

The formula for calculating mode for such grouped datasets is:
Mode = Z = L + (f1−f0) ÷ (2f1−f0−f2) × h
Where,
L = the lowest value of the modal class
h = the size of the class interval
f1 = the frequency of the modal class
f0 = the frequency of the class interval present before modal class
f2 = the frequency of the class interval present after model class
Some Important Terms for Mode of Grouped Data

Modal class: In grouped data, the group of numbers of the ‘interval’ that contains the mode or the greatest number of times a value occurs is called the modal class.

Class interval: Grouped datasets themselves can be of further two forms. One is where data is grouped such that there are intervals in between numbers, such as in the dataset table given above. Similarly, 2 to 10 is also an example of an interval.

However, if there are no intervals, data is grouped into two columns or rows where one contains observed, independent variables and the other contains frequencies or dependent variables corresponding to the observations.

For instance, suppose 10 children visit their street park 4 times and 5 children visit it 2 times. In such grouped data (with no intervals), the number of children is the independent variable, and their corresponding frequencies are the dependent variables.

The lower limit of the modal class: Since a class interval has two values—one where the interval begins and ends at—the lower or the starting one is called, the lower limit of that class interval.
Examples to Calculate Mode for Grouped Data

First, let’s consider grouped data with a class interval. Taking the same values as the one mentioned above,

Class levels (intervals) No. of students who obtained 99 marks in the final exam (frequencies)
2 – 4 3
4 – 6 4
6 – 8 2
8 – 10 1

and substituting the values in the formula below:

Mode = Z = L + (f1−f0) ÷ (2f1−f0−f2) × h

Where, 

  • L = 4
  • h = 2
  • f1 = 4
  • f0 = 3
  • f2 = 2

Mode = Z = 4+(4-3) ÷ (2 × 4-3-2) ×2

After performing the desired calculations, the mode comes out to be 4.67.

Similarly, the mode for the following grouped dataset is 10, for it’s the highest frequency occurring in all the 3 groups.

No. of children taking the bus to school Frequency
24 4
13 10
2 9

Where is Mode used?

Mode is commonly used alongside mean and median in statistics, mathematics, finance, accounting, academic research, engineering, software designing, and many other related fields and sub-fields. 
Is calculating the mode important? Why or why not?

The mode gives an idea about a dataset’s frequency; how often a certain value occurs. That further sheds light on that value’s importance, especially in research (explained below). 

1. In daily life, the mode is best applicable to situations where an average of a couple of datasets is to be compared with a single value. Mean and median can misrepresent such data, so using mode for the comparison is a better option.

2. Another very important reason to calculate mode is that it is one of the best statistical representations of nominal data. Qualitative things like a person’s gender, race, color, etc.—things that can’t otherwise be quantitatively measured, defined, or represented—comprise what is called nominal data.

On the other hand, quantitative values such as one’s height, weight, positions like first, second, third, etc. are all examples of ordinal data. In this type of data, an order is being followed, whereas it’s lacking in the case of nominal data.

Therefore, in research, the mode is best used to describe qualitative features of the research population or sample. Mean and median do not apply in such situations. For instance, research can sum up the frequency of males versus that of females using mode.

  1. Mode is not affected by the height of class intervals (for grouped data) nor the overall number of values in a dataset.

2. It’s easier to identify (for ungrouped data) than mean or median is.

3. Most importantly, the model allows for the creation of open-ended frequency distribution tables. Generally present in grouped data, intervals that are ‘open-ended’ mention something like ‘less than’ or ‘smaller’ than against their numbers. For instance, students who weigh 56kg or less, having a frequency of 10. This is a simple example of an open-ended frequency table.

When is mode NOT calculated?

The singular main situation where mode is not needed to be calculated is when either the data is ungrouped and fairly distributed, or when there are modes of a given dataset that’s also evenly distributed.

Yes, it can. When the dataset has one mode (one value that is being repeated the greatest number of times), it’s called a monomodal class. In the case of two, three, or more than three modes, it’s called a bimodal, trimodal and multimodal class, respectively.

Yes, two different values can have the same frequency. If that is the case, both are mentioned, along with the word ‘bimodal’ in front of ‘mode = Z =.’

Mode is generally denoted by the letter Z or read as Mo (for mode).

Yes, certainly. When such a dataset occurs, ‘no mode’ is written after ‘mode = Z =.’

Simply put, the mean is the average of a dataset divided by a total number of values within the set. Median is the central value of a dataset. And mode is the frequency of a dataset or the value which is repeated the most in the dataset.

About Alvin Nicolas

Nicolas has a master's degree in literature and a PhD degree in statistics. He loves to write, cook and run. Nicolas is passionate about helping students at all levels.